On the evaluation of highly oscillatory integrals with high frequency

We study accurate and stable approximations for highly oscillatory integrals (HOIs) which are demanding in computational engineering, especially for high frequency. In literature, the Levin method with radial basis functions (RBFs) is considered for better accuracy instead of stability of the algorithm. In this work, we demonstrate an accurate and stable Levin method for a class of RBFs. In addition, hybrid functions are coupled with Levin method to compute HOIs in case of stationary point, which returns a stable splitting procedure. Furthermore, error analysis of Levin method in the context of Gaussian RBFs is obtained. Although our error analysis is not significantly improved, however, we present a stable algorithm for evaluation of HOIs. The methods are implemented for both stationary and without stationary point for low and high frequencies. In each case, we address accuracy and stability of the algorithm which is reflected in numerical experiments. The obtained results are compared with well-known methods in the literature. Also, the numerical examples show that the proposed methods are highly accurate even for high frequency.